cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A122510 Array T(d,n) = number of integer lattice points inside the d-dimensional hypersphere of radius sqrt(n), read by ascending antidiagonals.

Original entry on oeis.org

1, 1, 3, 1, 5, 3, 1, 7, 9, 3, 1, 9, 19, 9, 5, 1, 11, 33, 27, 13, 5, 1, 13, 51, 65, 33, 21, 5, 1, 15, 73, 131, 89, 57, 21, 5, 1, 17, 99, 233, 221, 137, 81, 21, 5, 1, 19, 129, 379, 485, 333, 233, 81, 25, 7, 1, 21, 163, 577, 953, 797, 573, 297, 93, 29, 7, 1, 23, 201, 835, 1713, 1793
Offset: 1

Views

Author

R. J. Mathar, Oct 29 2006, Oct 31 2006

Keywords

Comments

Number of solutions to sum_(i=1,..,d) x[i]^2 <= n, x[i] in Z.

Examples

			T(2,2)=9 counts 1 pair (0,0) with sum 0, 4 pairs (-1,0),(1,0),(0,-1),(0,1) with sum 1 and 4 pairs (-1,-1),(-1,1),(1,1),(1,-1) with sum 2.
Array T(d,n) with rows d=1,2,3... and columns n=0,1,2,3.. reads
  1  3   3    3    5     5     5     5      5      7      7
  1  5   9    9   13    21    21    21     25     29     37
  1  7  19   27   33    57    81    81     93    123    147
  1  9  33   65   89   137   233   297    321    425    569
  1 11  51  131  221   333   573   893   1093   1343   1903
  1 13  73  233  485   797  1341  2301   3321   4197   5757
  1 15  99  379  953  1793  3081  5449   8893  12435  16859
  1 17 129  577 1713  3729  6865 12369  21697  33809  47921
  1 19 163  835 2869  7189 14581 27253  49861  84663 129303
  1 21 201 1161 4541 12965 29285 58085 110105 198765 327829

Links

Crossrefs

Rows d=1..10 give A001650, A057655, A117609, A046895, A175360, A175361, A341396, A341397, A341398, A341399.
Cf. A005408 (column 1), A058331 (column 2), A161712 (column 3), A055426 (column 4), A055427 (column 9)

Programs

  • Maple
    T := proc(d,n) local i,cnts ; cnts := 0 ; for i from -trunc(sqrt(n)) to trunc(sqrt(n)) do if n-i^2 >= 0 then if d > 1 then cnts := cnts+T(d-1,n-i^2) ; else cnts := cnts+1 ; fi ; fi ; od ; RETURN(cnts) ; end: for diag from 1 to 14 do for n from 0 to diag-1 do d := diag-n ; printf("%d,",T(d,n)) ; od ; od;
  • Mathematica
    t[d_, n_] := t[d, n] = t[d, n-1] + SquaresR[d, n]; t[d_, 0] = 1; Table[t[d-n, n], {d, 1, 12}, {n, 0, d-1}] // Flatten (* Jean-François Alcover, Jun 13 2013 *)

Formula

Recurrence along rows: T(d,n)=T(d,n-1)+A122141(d,n) for n>=1; T(d,n)=sum_{i=0..n} A122141(d,i). Recurrence along columns: cf. A123937.

A341427 Number of positive solutions to (x_1)^2 + (x_2)^2 + ... + (x_8)^2 <= n^2.

Original entry on oeis.org

1, 45, 767, 4452, 21178, 74452, 224313, 586035, 1387583, 2999430, 6102276, 11656386, 21282969, 37159993, 62687904, 102213426, 162345824, 251064745, 379922217, 562833191, 819351646, 1171991382, 1651937498, 2294227971, 3147090871, 4263499419
Offset: 3

Views

Author

Ilya Gutkovskiy, Feb 11 2021

Keywords

Links

Crossrefs

Cf. A000122, A001182, A055407, A055414, A253663, A302995, A340915, A341397, A341403, A341423, A341424, A341425, A341426, A341428, A341429.

Programs

  • Maple
    b:= proc(n, k) option remember; `if`(k=0, 1, `if`(n=0, 0,
      add((s->`if`(s>n, 0, b(n-s, k-1)))(j^2), j=1..isqrt(n))))
    end:
a:= n-> b(n^2, 8):
seq(a(n), n=3..28);  # Alois P. Heinz, Feb 11 2021
  • Mathematica
    Table[SeriesCoefficient[(EllipticTheta[3, 0, x] - 1)^8/(256 (1 - x)), {x, 0, n^2}], {n, 3, 28}]

Formula

a(n) is the coefficient of x^(n^2) in expansion of (theta_3(x) - 1)^8 / (256 * (1 - x)).

A341396 Number of integer solutions to (x_1)^2 + (x_2)^2 + ... + (x_7)^2 <= n.

Original entry on oeis.org

1, 15, 99, 379, 953, 1793, 3081, 5449, 8893, 12435, 16859, 24419, 33659, 42115, 53203, 69779, 88273, 106081, 125821, 153541, 187981, 217437, 248741, 298469, 351277, 394691, 446939, 515259, 589307, 657683, 728803, 828259, 939223, 1029159, 1124023, 1260103
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 10 2021

Keywords

Comments

Partial sums of A008451.

Links

Crossrefs

Cf. A000122, A001650, A008451, A046895, A055406, A055413, A057655, A117609, A122510, A175360, A175361, A302860, A341397, A341398, A341399.

Programs

  • Maple
    b:= proc(n, k) option remember; `if`(n=0, 1, `if`(n<0 or k<1, 0,
      b(n, k-1)+2*add(b(n-j^2, k-1), j=1..isqrt(n))))
    end:
a:= proc(n) option remember; b(n, 7)+`if`(n>0, a(n-1), 0) end:
seq(a(n), n=0..35);  # Alois P. Heinz, Feb 10 2021
  • Mathematica
    nmax = 35; CoefficientList[Series[EllipticTheta[3, 0, x]^7/(1 - x), {x, 0, nmax}], x]
Table[SquaresR[7, n], {n, 0, 35}] // Accumulate
  • PARI
    my(q='q+O('q^(55))); Vec((eta(q^2)^5/(eta(q)^2*eta(q^4)^2))^7/(1-q)) \\ Joerg Arndt, Jun 21 2024

Formula

G.f.: theta_3(x)^7 / (1 - x).
a(n^2) = A055413(n).

A341398 Number of integer solutions to (x_1)^2 + (x_2)^2 + ... + (x_9)^2 <= n.

Original entry on oeis.org

1, 19, 163, 835, 2869, 7189, 14581, 27253, 49861, 84663, 129303, 190071, 284055, 409335, 550455, 732855, 995241, 1312617, 1656153, 2077497, 2634777, 3300057, 4003641, 4804281, 5872665, 7129227, 8363307, 9784491, 11635755, 13670475, 15727755, 18066315, 20950491
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 10 2021

Keywords

Comments

Partial sums of A008452.

Links

Crossrefs

Cf. A000122, A001650, A008452, A046895, A055408, A055415, A057655, A117609, A122510, A175360, A175361, A302860, A341396, A341397, A341399.

Programs

  • Maple
    b:= proc(n, k) option remember; `if`(n=0, 1, `if`(n<0 or k<1, 0,
      b(n, k-1)+2*add(b(n-j^2, k-1), j=1..isqrt(n))))
    end:
a:= proc(n) option remember; b(n, 9)+`if`(n>0, a(n-1), 0) end:
seq(a(n), n=0..32);  # Alois P. Heinz, Feb 10 2021
  • Mathematica
    nmax = 32; CoefficientList[Series[EllipticTheta[3, 0, x]^9/(1 - x), {x, 0, nmax}], x]
Table[SquaresR[9, n], {n, 0, 32}] // Accumulate

Formula

G.f.: theta_3(x)^9 / (1 - x).
a(n^2) = A055415(n).

A341399 Number of integer solutions to (x_1)^2 + (x_2)^2 + ... + (x_10)^2 <= n.

Original entry on oeis.org

1, 21, 201, 1161, 4541, 12965, 29285, 58085, 110105, 198765, 327829, 503509, 765589, 1152509, 1642109, 2243069, 3083569, 4221529, 5551949, 7115789, 9166133, 11777333, 14763893, 18121973, 22316213, 27634481, 33512921, 39812441, 47674841, 57294401, 67510721, 78592961
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 10 2021

Keywords

Comments

Partial sums of A000144.

Links

Crossrefs

Cf. A000122, A000144, A001650, A046895, A055409, A055416, A057655, A117609, A122510, A175360, A175361, A302860, A341396, A341397, A341398.

Programs

  • Maple
    b:= proc(n, k) option remember; `if`(n=0, 1, `if`(n<0 or k<1, 0,
      b(n, k-1)+2*add(b(n-j^2, k-1), j=1..isqrt(n))))
    end:
a:= proc(n) option remember; b(n, 10)+`if`(n>0, a(n-1), 0) end:
seq(a(n), n=0..31);  # Alois P. Heinz, Feb 10 2021
  • Mathematica
    nmax = 31; CoefficientList[Series[EllipticTheta[3, 0, x]^10/(1 - x), {x, 0, nmax}], x]
Table[SquaresR[10, n], {n, 0, 31}] // Accumulate

Formula

G.f.: theta_3(x)^10 / (1 - x).
a(n^2) = A055416(n).

A373896 Number of lattice points inside or on the 8-dimensional hypersphere x_1^2 + x_2^2 + x_3^2 + x_4^2 + x_5^2 + x_6^2 + x_7^2 + x_8^2 = 10^n.

Original entry on oeis.org

17, 47921, 415055025, 4068011664081, 40596481219349025, 405880555110153633585, 4058721509888208894731345, 40587130610718907618215585345, 405871222004868007901459647593809, 4058712135741827985063748936303681217
Offset: 0

Views

Author

Seiichi Manyama, Jun 21 2024

Keywords

Crossrefs

Cf. A068785, A373881, A373882, A373883, A373884, A373885.
Cf. A008457, A341397.

Programs

  • PARI
    a008457(n) = sumdiv(n, d, (-1)^(n-d)*d^3);
a341397(n) = 1+16*sum(k=1, n, a008457(k));
a(n) = a341397(10^n);

Formula

a(n) = A341397(10^n).
a(n) == 1 (mod 16).
Showing 1-6 of 6 results.

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